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<Title>Gravity</Title>
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<h1>Newton's law of gravitation</h1>
<p>
Half a century after Johannes Kepler discovered the laws according to which planets move with respect to the sun, Sir Isaac Newton (1642-1727) formulated his law of gravitation, which is fundamental to the understanding of the dynamics of orbital motion, and from which Kepler's laws can be directly derived.
<p>
The law of gravitation states that an <i>attracting force</i> exists between two bodies, which is proportional to their <i>inverse square distance</i>. The magnitude of the force also depends on a physical property of the two bodies, called their <i>mass</i>.
<p>
Mathematically, the law of gravitation can be stated like this:
<blockquote>
<img src="gravitylaw.gif">
</blockquote>
where <font class="math"><i>F<sub>g</sub></i></font> is the magnitude of the gravitational force acting on each body.
<font class="math"><i>m</i><sub>1</sub></font> and <font class="math"><i>m</i><sub>2</sub></font> are the masses of body 1 and 2, and <font class="math"><i>r</i></font> is the distance between them.
<p>
<font class="math"><i>G</i></font> is a universal constant, called the <i>gravitational constant</i>. Its value, within the current measurement uncertainty, is given by
<blockquote>
<img src="gravityconst.gif">
</blockquote>
(Newtons times metres squared per kilograms squared)
<p>
So far we have only considered the magnitude of the gravitational force, but forces also have a <i>direction</i>. In mathematics, quantities with a magnitude and direction are described by <i>vectors</i>. We can state the law of gravitation in vector form in the following way:
<blockquote>
<img src="gravityvec.gif">
</blockquote>
This is similar to the first equation, but now the object on the left side is a vector (marked by the bold typeface). On the right side, the expression <font class="math"><b>r</b><sub>12</sub>/|<b>r</b><sub>12</sub>|</font> denotes a <i> unit vector</i> (that is, a vector of length 1) pointing from body 1 in the direction of body 2.
To understand this, note that the numerator, <font class="math"><b>r</b><sub>12</sub></font>, is the <i>distance vector</i> between body 1 and 2, that is, a vector with its origin at the position of body 1, and its tip at the position of body 2. The denominator, <font class="math">|<b>r</b><sub>12</sub>|</font>, (the vector enclosed in vertical bars) means the <i>norm</i> or <i>length</i> of the vector. Dividing a vector by its length always results in a vector of length 1, pointing in the same direction as the original vector.
<p>
<img src="gforce.gif" align=right hspace=10 vspace=10>
The equation above describes the force acting on body 2 (hence the subscript 2 was added to the force symbol on the left hand side). Note the minus sign on the right, which indicates that the force vector is pointing from body 2 towards body 1.
<p>
The force acting on body 1 has the same magnitude, but opposite direction (according to Newton's third law of actio = reactio):
<blockquote>
<img src="gravityvec2.gif">
</blockquote>
So in a nutshell:
<ul>
<li>Two bodies of mass <font class="math"><i>m</i><sub>1</sub></font> and <font class="math"><i>m</i><sub>2</sub></font> at distance <font class="math"><i>r</i></font> will experience an attracting force of equal magnitude but opposite direction.
<li>The magnitude of the force is proportional to the product of the masses of the two bodies, and inversely proportional to the square of their distance.
<li>The proportionality constant is given by the gravitational constant <font class="math"><i>G</i></font>.
</ul>
<p>
<table cols=2 width=100%>
<tr><td width=50% align=center></td>
<td width=50% align=center><a href="Gravity2.htm">Gravitational potential</a>&nbsp;&gt;&gt;</td></tr>
</table>

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